This paper presents a method to improve the fatigue strength of 2D periodic cellular materials under a fully-reversed loading condition. For a given cell topology, the shape of the unit cell is synthesized to minimize any stress concentration caused by discontinuities in the cell geometry. We propose to reduce abrupt geometric changes emerging in the periodic microstructure through the synthesis of a cell shape defined by curved boundaries with continuous curvature, i.e. G2-continous curves. The bending moments caused by curved cell elements are reduced by minimizing the curvature of G2-continuous cell elements so as to make them as straight as possible. The asymptotic homogenization technique is used to obtain the homogenized stiffness matrix and the fatigue strength of the synthesized cellular material. The proposed methodology is applied to synthesize a unit cell topology described by smooth boundary curves. Numeric simulations are performed to compare the performance of the synthesized cellular solid with that of common two dimensional lattice materials having hexagonal, circular, square, and Kagome shape of the unit cell. The results show that the methodology enables to obtain a cellular material with improved fatigue strength. Finally, a parametric study is performed to examine the effect of different geometric parameters on the performance of the proposed cellular geometries.

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